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Moduli Spaces: New Directions

模空间：新方向

基本信息

批准号：
1701704
负责人：
Evgueni Tevelev
金额：
$ 20万
依托单位：
University of Massachusetts Amherst
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-15 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1701704&HistoricalAwards=false
关键词：
Moduli Spaces New Directions

项目摘要

Algebraic geometry studies varieties: shapes defined by systems of polynomial equations. Polynomial constraints on parameters are quite common both in mathematics (for example, Diophantine equations in number theory, such as the famous Fermat equation) and in broader applications ranging from physics, where algebraic varieties are used to describe various configuration spaces (and even the shape of the universe!), to engineering and algebraic statistics. The inherent rigidity of polynomials, especially compared to more flexible transformations allowed in topology and differential geometry, makes possible to apply a wide range of techniques coming from algebra (the study of rings and modules) and even number theory, for example reduction modulo prime numbers. One of the most basic problems in algebraic geometry is to learn how to classify and distinguish geometric properties of algebraic varieties. In addition to discrete parameters, mostly of topological nature, algebraic varieties also have continuous parameters called "moduli". For example, Riemann surfaces (such as a sphere or a torus) are classified topologically by the number of holes (the genus). It was known already to Riemann that they also have 3g-3 continuous complex parameters (g is the genus). Quite remarkably, these parameters can be interpreted as coordinates on the "moduli space": the master space that encodes all possible geometric structures of a given topological shape. The PI is an expert on moduli spaces of low-dimensional varieties (curves and surfaces). The project will focus on developing new techniques and approaches to their study. The first approach is based on convex geometry, which studies shapes defined by linear inequalities (such as polygons and polyhedra). An idea that goes back to Newton is to capture asymptotic behavior of a distinguished system of functions to construct a polyhedral approximation of a curved geometry. Another approach is linearization. Derived categories linearize geometries just like tangent lines approximate graphs of functions. The PI aims to understand derived categories of several moduli spaces. Finally, moduli spaces can be studied using number theory by reducing them modulo prime numbers. To this end, moduli spaces have to be equipped with extra data to parametrize complex and finite geometries simultaneously. In addition to advancing algebraic geometry, the proposed research program contributes to the development of a diverse, globally competitive STEM workforce through recruiting, training, and supervising of graduate and undergraduate students and providing resources for their research, teaching, and professional development. In particular, the project includes three problems specifically tailored for REU (Research Experience for Undergraduates) as well as several PhD thesis problems. The PI will also continue to develop graduate and undergraduate courses, including courses designed to introduce a broader public to mathematics and science.The PI will work on several fundamental problems in algebraic geometry with a focus on compact moduli spaces of local systems, curves and surfaces. The PI has previously developed applications of classical "commutative" tropicalization of subvarieties in algebraic tori to moduli spaces. Recently, classical tropical geometry was extended by the PI and his graduate student Vogiannou to a "non-commutative" setting and the PI proposes to use this generalization to construct a geometrically meaningful compactification of the moduli space of local systems on a punctured Riemann sphere. In collaboration with Castravet, the PI intends to verify a conjecture of Orlov and Kuznetsov on derived category of the moduli space of stable rational curves (and related spaces) and to apply this description to find its non-commutative deformations. The PI also intends to use the theory of windows as developed by Halpern-Leistner to solve some open problems about derived categories of toric and log Fano varieties. In collaboration with Freixas i Montplet, the PI proposes to systematically develop Arakelov geometry of the moduli space of curves and to apply it to questions in number theory. The PI proposes several new projects on moduli of surfaces in collaboration with Urzua, including constructions of moduli spaces of surfaces with QHD singularities (admitting smoothings with rational homology disc Milnor fiber) and families of antiflips of total spaces of deformations of cyclic quotient singularities in the direction of answering a question of Kollar. In addition there are three projects tailored for REUs: Seshadri constants on toric and elliptic ruled surfaces, and equations of log canonical compactifications of complements of hyperplane arrangements.

代数几何研究多样性：由多项式方程组定义的形状。参数的多项式约束在数学（例如，数论中的丢番图方程，例如著名的费马方程）和更广泛的应用中都很常见，从物理学（代数簇用于描述各种配置空间（甚至宇宙的形状！））到工程和代数统计。多项式固有的刚性，特别是与拓扑和微分几何中允许的更灵活的变换相比，使得应用来自代数（环和模的研究）和偶数论的广泛技术成为可能，例如约简模素数。代数几何最基本的问题之一是学习如何分类和区分代数簇的几何性质。除了大多数具有拓扑性质的离散参数之外，代数簇还具有称为“模”的连续参数。例如，黎曼曲面（例如球体或环面）根据孔的数量（属）进行拓扑分类。黎曼已经知道它们也有 3g-3 个连续复参数（g 是亏格）。值得注意的是，这些参数可以解释为“模空间”上的坐标：对给定拓扑形状的所有可能几何结构进行编码的主空间。 PI 是低维变量（曲线和曲面）模空间方面的专家。该项目将重点开发新的研究技术和方法。第一种方法基于凸几何，它研究由线性不等式定义的形状（例如多边形和多面体）。一个可以追溯到牛顿的想法是捕捉一个杰出的函数系统的渐近行为来构造弯曲几何的多面体近似。另一种方法是线性化。派生类别将几何图形线性化，就像切线近似函数图一样。 PI 旨在理解几个模空间的派生类别。最后，可以使用数论通过将模素数约简来研究模空间。为此，模空间必须配备额外的数据来同时参数化复杂和有限的几何形状。除了推进代数几何之外，拟议的研究计划还通过招募、培训和监督研究生和本科生以及为他们的研究、教学和专业发展提供资源，有助于培养多元化、具有全球竞争力的 STEM 劳动力。特别是，该项目包括专门为REU（本科生研究经验）量身定制的三个问题以及几个博士论文问题。 PI 还将继续开发研究生和本科生课程，包括旨在向更广泛的公众介绍数学和科学的课程。PI 将致力于解决代数几何中的几个基本问题，重点是局部系统、曲线和曲面的紧模空间。 PI 之前已经开发了代数环面中子品种的经典“交换”热带化到模空间的应用。最近，PI和他的研究生Vogiannou将经典热带几何扩展到“非交换”环境，并且PI建议使用这种推广来构造穿孔黎曼球上局部系统模空间的几何上有意义的紧致化。 PI 打算与 Castravet 合作，验证奥尔洛夫和库兹涅佐夫关于稳定有理曲线模空间（及相关空间）的派生范畴的猜想，并应用此描述来找到其非交换变形。 PI 还打算使用 Halpern-Leistner 开发的窗口理论来解决有关 toric 和 log Fano 变体的派生类别的一些开放问题。 PI 与 Freixas i Montplet 合作，建议系统地开发曲线模空间的 Arakelov 几何，并将其应用于数论问题。 PI 与 Urzua 合作提出了几个关于曲面模量的新项目，包括构造具有 QHD 奇点的曲面模空间（允许使用有理同源盘 Milnor 纤维进行平滑）以及在回答 Kollar 问题的方向上循环商奇点变形总空间的反翻转族。此外，还有三个专为 REU 定制的项目：环曲面和椭圆直纹面上的 Seshadri 常数，以及超平面排列补集的对数正则紧化方程。